Loophole-Free Bell Tests

Loophole-free Bell tests are experiments designed to violate a Bell inequality while closing the major implementation loopholes that affected earlier tests. The phrase is technical. It usually means that the experiment closes the detection and locality loopholes in one run, with a predefined statistical analysis, under stated assumptions.

It does not mean that the experiment is assumption-free. Bell tests still require assumptions about setting independence, event definitions, statistical inference, and the physical relevance of the implemented measurement scenario. The achievement is nevertheless deep: modern experiments rule out broad classes of local hidden-variable explanations without relying on the fair-sampling and static-setting assumptions that earlier tests needed.

For the general survey, see Bell Inequality Experiments. For the historical bridge, see Aspect Experiments.

What Loophole-Free Means

A Bell inequality is derived for an idealized scenario: trial settings are chosen, outcomes are assigned, correlations are estimated, and the resulting Bell parameter is compared with a local hidden-variable bound. In the CHSH case,

$$\lvert S\rvert\le2$$

for local hidden-variable models satisfying the CHSH assumptions.

A loophole-free experiment tries to make the laboratory trial structure close enough to the ideal structure that a local hidden-variable model cannot exploit ordinary implementation gaps. The two traditional gaps are:

• missed events, which create the detection loophole;
• causal contact between setting choices and distant outcomes, which creates the locality loophole.

Modern experiments also predefine timing windows, setting-generation procedures, and statistical tests. This is why a mature Bell test is not just an optical alignment problem. It is a spacetime, detection, and inference problem.

Detection Loophole

The detection loophole appears when only a subset of eligible trials produces registered outcomes. If the detected subset depends on hidden variables and settings, the measured correlations may violate a Bell inequality even though the full underlying ensemble would not.

Detection efficiency concerns the probability of actually registering the system in an eligible trial. It is not the fraction of trials to which the analysis assigns a binary value: assigning every nondetection the value $+1$ gives an outcome on every trial without improving the detector.

Dropping nondetections and keeping only coincidences generally requires a fair-sampling assumption. An analysis that includes every eligible trial can instead assign a specified outcome to nondetections or use a Bell inequality that treats them explicitly. Low efficiency then makes a violation harder to obtain, rather than invalidating the local bound.

Event-ready heralding defines eligible trials before the measurement settings and outcomes used in the test. A herald identifies a successful preparation attempt; it does not guarantee a perfect entangled state on every attempt. Closing detection additionally requires sufficiently efficient local readout and retention of all heralded trials under the predefined rules. Heralding alone is not a cure for inefficient final detectors.

A local selection model

The danger of discarding missed detections can be calculated without a quantum state. Pearle’s 1970 hidden-variable model established the importance of this mechanism. The following finite example illustrates it for four setting pairs; it is not a model of any particular experiment.

Label the independently chosen settings $x,y\in\{0,1\}$. Before those choices, a source draws three independent variables: two fair bits $x_0,y_0$ and a fair sign $r=\pm1$. It sends the shared record $\lambda=(x_0,y_0,r)$ to both stations. Its distribution is independent of the actual settings. The detectors follow local rules:

• Alice detects only if $x=x_0$ and records $A=r$.
• Bob detects only if $y=y_0$ and records $B=r(-1)^{x_0y_0}$.

Each rule uses only the local setting and the shared record. Neither station needs the distant setting. For every setting pair, each detector fires on half the trials and both fire on one quarter. Within the coincidences, both signs occur equally often at either station. Nevertheless, keeping only coincidences selects $x_0=x$ and $y_0=y$, so

$$E_{\mathrm{coinc}}(x,y)=(-1)^{xy}, \qquad S_{\mathrm{coinc}}=1+1+1-(-1)=4.$$

This apparent violation even exceeds the quantum CHSH bound. It comes from using a different hidden-variable subensemble for each setting pair. Constant detection rates and balanced selected marginals do not establish fair sampling. The original source distribution remains setting independent; the distribution conditioned on both detections does not. Including every trial with a fixed nondetection outcome restores an ordinary local model, as Exercise 2 checks.

Locality Loophole

The locality loophole concerns whether information about one side’s setting or outcome could influence the other side’s outcome through an ordinary signal moving at or below the speed of light.

If the two stations are separated by distance $L$, then a rough timing requirement is

$$\lvert\Delta t\rvert < \frac{L}{c},$$

where $\Delta t$ is the time difference between a setting choice on one side and an outcome event on the other. Each station must satisfy the relevant remote-setting constraint, including the duration and uncertainty of its choice and readout. Real analyses use a more careful spacetime diagram, but the principle is this: the relevant events should be spacelike separated.

Aspect’s time-varying analyzer experiment was an important step toward this requirement. Modern tests go further by using fast setting generation, better synchronization, larger separations, and careful event timing. A locality-closed Bell test must show not only a violation, but also that the settings and outcomes were arranged so that ordinary causal communication cannot explain the observed correlations.

Freedom-of-Choice Assumptions

Bell tests also assume that the hidden variables associated with the systems are statistically independent of the later setting choices. This is often called measurement independence or setting independence.

Experiments can make this assumption physically plausible by using fast random setting generators placed near the measurement stations, by shielding devices, and by using distant physical processes to choose settings. Cosmic Bell tests and human-choice experiments address this assumption in different ways. They push possible common-cause explanations farther into the past or into more contrived forms.

They do not eliminate the logical possibility of superdeterministic correlations between all setting choices and hidden variables. That possibility is not a normal experimental loophole in the same sense as detector inefficiency. It is a different kind of assumption about the relation between experimental choices and the variables being tested.

2010s Milestones

The year 2015 marked a turning point. Three experiments reported Bell violations while closing the major detection and locality loopholes in the same test.

Hensen and collaborators used electron spins associated with nitrogen-vacancy centers in diamond separated by about $1.3$ km. An event-ready entanglement-swapping signal heralded valid trials before the final spin measurements. Efficient spin readout avoided fair sampling, while fast random settings and the separation supplied the locality conditions. The experiment reported 245 trials and a memory-tolerant null-hypothesis p-value of 0.039; a Bell parameter above the bound alone is not a complete statistical result.

Giustina and collaborators and Shalm and collaborators performed high-efficiency photonic Bell tests. These experiments used improved entangled-photon sources, high-efficiency detectors, fast setting choices, and statistical analyses designed for loophole-free claims. They addressed the historical weakness of photonic tests: losses that could otherwise reopen the detection loophole.

Later work broadened the landscape. Event-ready tests with atoms, cosmic setting-choice tests, large multi-laboratory human-choice tests, and superconducting-circuit Bell tests all explored different assumptions and platforms. These experiments address different conditions: using cosmic or human settings does not by itself close the detection loophole. Their individual protocols must be examined before combining their conclusions.

Device-Independent Quantum Information

Loophole-free Bell tests matter beyond foundations because Bell violation can be used as an operational certificate. In device-independent protocols, one tries to certify a property using observed input-output statistics rather than a trusted microscopic model of the devices.

Examples include:

• randomness certification;
• device-independent quantum key distribution;
• self-testing of states and measurements;
• entanglement certification under minimal trust assumptions.

The word “device-independent” also has assumptions. The devices must be isolated from unwanted communication, the setting choices must have appropriate independence, and the statistical analysis must account for finite data. Loophole-free Bell tests supply the experimental discipline needed for those protocols.

Careful Interpretation

Modern loophole-free Bell tests strongly rule out local hidden-variable explanations satisfying the tested assumptions. They support the quantum prediction that suitable entangled systems violate Bell inequalities while preserving no-signaling at the level of local marginal statistics.

They do not prove controllable faster-than-light communication. They do not choose a unique interpretation of quantum mechanics. They do not eliminate every logically possible hidden-variable theory; nonlocal theories and theories rejecting setting independence are outside the excluded class.

Exercises

1. A Bell test has two stations separated by $1.2$ km. Estimate $L/c$ and explain why this timescale matters.

Solution

Using $c\approx3.0\times10^8\,\mathrm{m/s}$,

$$\frac{L}{c} = \frac{1.2\times10^3\,\mathrm{m}} {3.0\times10^8\,\mathrm{m/s}} \approx 4.0\,\mu\mathrm s.$$

If a setting choice on one side and the relevant outcome event on the other side occur within a shorter interval than this light-travel time, they can be arranged to be spacelike separated. That helps close the locality loophole.

2. In the local selection model above, assign the outcome $+1$ to every nondetection and retain all trials. Find the four correlations and their CHSH combination.

Solution

For fixed $x,y$, each of the four detection patterns has probability $1/4$. When both detectors fire, the product is $(-1)^{xy}$. When just one fires, its fair sign averages to zero; when neither fires, the assigned product is $+1$. Thus

$$E_{\mathrm{all}}(x,y)=\frac{(-1)^{xy}+1}{4}.$$

In the order $(0,0),(0,1),(1,0),(1,1)$ the correlations are $1/2,1/2,1/2,0$, giving $S_{\mathrm{all}}=3/2\le2$. Assigning an outcome has not improved either detector. It has removed the biased selection from the comparison with the local bound.

3. Why does a cosmic setting-choice experiment not eliminate every possible freedom-of-choice concern?

Solution

Cosmic setting choices push possible common causes far into the past by using distant astronomical events to help choose measurement settings. However, a sufficiently extreme superdeterministic model could still posit correlations between those astronomical events, the source variables, and the experimenters’ choices. Such a possibility is not removed by any ordinary finite experiment; it is a different kind of assumption.

4. State the careful conclusion of loophole-free Bell tests in one sentence.

Solution

Modern loophole-free Bell tests show that nature violates Bell inequalities in experiments that close the major detection and locality loopholes under explicit assumptions about setting independence, event definitions, and statistical analysis.

References

• J. S. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed., Cambridge University Press, 2004.
• The BIG Bell Test Collaboration, “Challenging local realism with human choices,” Nature 557, 212-216, 2018, DOI: 10.1038/s41586-018-0085-3.
• N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Reviews of Modern Physics 86, 419-478, 2014, DOI: 10.1103/RevModPhys.86.419.
• M. Giustina et al., “Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons,” Physical Review Letters 115, 250401, 2015, DOI: 10.1103/PhysRevLett.115.250401.
• B. Hensen et al., “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526, 682-686, 2015, DOI: 10.1038/nature15759.
• P. M. Pearle, “Hidden-Variable Example Based upon Data Rejection,” Physical Review D 2, 1418-1425, 1970, DOI: 10.1103/PhysRevD.2.1418.
• D. Rauch et al., “Cosmic Bell Test Using Random Measurement Settings from High-Redshift Quasars,” Physical Review Letters 121, 080403, 2018, DOI: 10.1103/PhysRevLett.121.080403.
• W. Rosenfeld et al., “Event-Ready Bell Test Using Entangled Atoms Simultaneously Closing Detection and Locality Loopholes,” Physical Review Letters 119, 010402, 2017, DOI: 10.1103/PhysRevLett.119.010402.
• L. K. Shalm et al., “Strong Loophole-Free Test of Local Realism,” Physical Review Letters 115, 250402, 2015, DOI: 10.1103/PhysRevLett.115.250402.
• S. Storz et al., “Loophole-free Bell inequality violation with superconducting circuits,” Nature 617, 265-270, 2023, DOI: 10.1038/s41586-023-05885-0.